Abstract

We introduce a new concept of second-order asymptotic directions of unbounded sets and apply it to establish global optimality conditions of nonlinear functions on unbounded sets, existence conditions for efficient points in a partially ordered space, and to extend Dieudonné's theorem on the closedness of the sum of two closed sets.

Author information

Contact details are reproduced from the original publication and may be historical.

Nicolas Hadjisavvas

Dept. of Product and Systems Design, University of the Aegean, 84100 Hermoupolis - Syros, Greece

nhad@aegean.gr

Dinh The Luc

Laboratoire ANLG EA 2151, University of Avignon, 33 Rue Louis Pasteur, 84000 Avignon, France

Keywords

  • First-order and second-order asymptotic directions; optimality conditions; efficient points

Mathematics Subject Classification

90C30, 90C29, 49J52

References

  1. S. Adly, D. Goeleven, M. Théra: Recession mappings and noncoercive variational inequalities, Nonlinear Anal. 26 (1996) 1573–1603.https://doi.org/10.1016/0362-546X(94)00364-NGoogle Scholar
  2. A. Agadi: Approximation des Ensembles en un Point et à l’Infini, Thèse de Doctorat, Université de Pau (1995).Google Scholar
  3. J.-P. Aubin, H. Frankowska: Set-Valued Analysis, Birkhäuser, Basel (1990).Google Scholar
  4. A. Auslender, M. Teboulle: Asymptotic Cones and Functions in Optimization and Variational Inequalities, Springer Monographs in Mathematics, Springer, New York (2003).Google Scholar
  5. G. Debreu: Theory of Value, Wiley, New York (1975).Google Scholar
  6. J.-P. Dedieu: Cône asymptote d’un ensemble non convexe. Application à l’optimisation, C. R. Acad. Sci., Paris, Sér. A 285 (1977) 501–503.Google Scholar
  7. J. Dieudonne: Sur la séparation des ensembles convexes, Math. Ann. 163 (1966) 1–3.https://doi.org/10.1007/BF02052480Google Scholar
  8. V. L. Klee: Asymptotes and projections of convex sets, Math. Scand. 8 (1960) 356–362.https://doi.org/10.7146/math.scand.a-10617Google Scholar
  9. D. T. Luc: Theory of Vector Optimization, Lect. Notes Econ. Math. Sci. 319, Springer, Berlin (1989).https://doi.org/10.1007/978-3-642-50280-4Google Scholar
  10. D. T. Luc: Recession cones and the domination property in vector optimization, Math. Program., Ser. A 49 (1990) 113–122.https://doi.org/10.1007/BF01588781Google Scholar
  11. D. T. Luc: Recession maps and applications, Optimization 27 (1993) 1–15.https://doi.org/10.1080/02331939308843868Google Scholar
  12. D. T. Luc: Recessively compact sets: uses and properties, Set-Valued Anal. 10 (2002) 15–35.https://doi.org/10.1023/A:1014458603461Google Scholar
  13. D. T. Luc, J. P. Penot: Convergence of asymptotic directions, Trans. Amer. Math. Soc. 353 (2001) 4095–4121.https://doi.org/10.1090/S0002-9947-01-02664-2Google Scholar
  14. D. T. Luc, M. Théra: Derivatives with support and applications, Math. Oper. Res. 19 (1994) 659–675.https://doi.org/10.1287/moor.19.3.659Google Scholar
  15. D. T. Luc, R. J.-B. Wets, Outer semicontinuity of positive hull mappings with application to semi-infinite and stochastic programming, SIAM J. Optim. 19 (2008) 700–713.https://doi.org/10.1137/070687505Google Scholar
  16. P. K. Monteiro, F. H. Page Jr, M. H. Wooders: Arbitrage, equilibrium, and gains from trade: A counterexample, J. Math. Econ. 28 (1997) 481–501.https://doi.org/10.1016/S0304-4068(96)00774-4Google Scholar
  17. J.-P. Penot: The cosmic Hausdorff topology, the bounded Hausdorff topology and continuity of polarity, Proc. Amer. Math. Soc. 113 (1991) 275–285.https://doi.org/10.1090/S0002-9939-1991-1068129-XGoogle Scholar
  18. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  19. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (1997).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  20. R. E. Steinitz: Bedingt konvergente Reihen und konvexe Systeme I, II, III, J. für Math. 143 (1913) 128–175; 144 (1914) 1– 40; 146 (1916) 1–52.https://doi.org/10.1515/crll.1914.144.1Google Scholar
  21. C. Zǎlinescu: Recession cones and asymptotically compact sets, J. Optim. Theory Appl. 77 (1993) 209–220.https://doi.org/10.1007/BF00940787Google Scholar

Suggested citation

N. Hadjisavvas, D. T. Luc. “Second-Order Asymptotic Directions of Unbounded Sets with Application to Optimization.” Journal of Convex Analysis 18 (2011), No. 1, 181–202. https://doi.org/10.68381/jca18010

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Published by Heldermann Verlag, 2011. Rights now held by Banach Press LLC.